This generalized golden ratio \(\mathcal {G}(m)\)  plays an important part in the sense that for \(q \in (1,\mathcal {G}(m))\)  every \(x \in (0,\frac{m}{q-1})\)  has uncountable \(q\) -expansions, and for \(q \in (\mathcal {G}(m),m+1]\)  there exists \(x \in (0,\frac{m}{q-1})\)  that has a unique \(q\) -expansion. Thus \(\mathcal {U}_q=\left\lbrace 0,\frac{m}{q-1}\right\rbrace \)  for all \(q\in (1,\mathcal {G}(m)]\)  (see, e.g., [1]}, [2]}).
In the whole paper, we always use \(\alpha (q)=(\alpha _i)\)  and \(\beta (q)=(\beta _i)\)  to denote the quasi-greedy and greedy \(q\) -expansions of 1 respectively. The following property of quasi-greedy expansion of 1 is related to Parry's work [3]} (see also [4]}, [5]}).
